I'll try to motivate and introduce the modern formulation of a topological quantum field theory (TQFT), inspired by Segal's and Atiyah's original axioms,
as a symmetric monoidal functor from a certain bordism category to the category of vector spaces. For motivation, I'll look at classical field theories in physics,
and explain how a topological quantum field theory can be viewed as a ``quantization'' of a certain classical model, called the non-linear sigma model.
If time permits, I'll discuss examples of TQFTs and their applications to other areas.

Feb 18

Joshua Seaton

Extrapolating factorials: the Gamma function, classical and p-adic

Graph the points \((n, n!)\) in the plane for positive integers \(n\); there are a continuum of curves that go through these points.
So why does the Gamma function -- among the infinitely-many other candidates -- deserve to be regarded as the 'factorial function'?
The Bohr-Mollerup-Artin Theorem tells us why, which reveals that a function satisfying a couple of mild 'factorial'-like properties necessarily
has to be the Gamma function. We first go through this brief, neat result.
Then, in the second part of this talk, we look at the p-adic side of things (we will introduce p-adic numbers gently here, I promise.)
In the ring of p-adic integers, the usual integers are dense. So -- in high contrast to the setting over the real numbers -- an integer-valued
function on the integers (if p-adically continuous) will lift uniquely to a function on the p-adic integers!
After some minor tweaking to the function that maps \(n\) to \(n!\), we will take a brief look at its (unique) p-adic extension, the p-adic Gamma function.

Feb 25

Linus Hamann

A journey throughelliptic curves

In this talk, we will begin with a classical overview of the definition of an elliptic curve and associated properties.
Afterwords, we will introduce some of the tools and terminology used in the algebraic geometry of smooth curves, including the Picard group, differentials, and Riemann-Roch.
The remainder of the talk will be dedicated to using this new terminology to illuminate various features of the classical view of elliptic curves and
thereby give insight into their importance.

March 4

Irit Huq-Kuruvilla

The combinatorialNullstellensatz

In this talk I will state and prove the Combinatorial Nullstellensatz and illustrate the power of the theorem by using it to give short,
beautiful solutions to many problems. The results we will prove include the Cauchy-Davenport theorem, a fundamental result in additive number theory,
and a result about hyperplane arrangements that was the 6th problem at the 2007 International Mathematical Olympiad. This talk presumes
little technical knowledge, and should be accessible to general audiences.

March 11

Karsten Gimre

PDEs andgeneral relativity

I will describe some advances made in hyperbolic partial differential equations in the last thirty years, with an eye towards application
in the dynamics of Einstein's field equations, in particular gravitational wave detection and formation of black holes.
It will not be assumed that the audience is familiar with either hyperbolic PDE or relativity.

March 18

Spring break

March 25

Sander Mack-Crane

Moonshine

Moonshine is a sporadic collection of mysterious connections between the algebraic world of finite groups and the number-theoretic world of modular functions.
We will first introduce these worlds and discuss their independent interest. Then we will examine the moonshine that connects them, starting with its discovery
and building up to some recent directions in the theory.

April 1

Willie Dong

Regularity of the Linearized Navier-Stokes Equations

In this talk, we will state the linearized Navier-Stokes equations, prove the existence and uniqueness of weak solutions, and use Lebesgue
elliptic estimates to derive regularity results; we will introduce tools from classical functional analysis (i.e. segment property, function
spaces, weak convergence) along the way.

April 8

Robbie Lyman

Geometric Group Theory

April 15

Matei IonitaNilay Kumar

Borel-Weil-Bottand Beilinson-Bernstein

We review the basics of the representation theory of complex semisimple Lie algebras via the theory of highest weights.
We make contact with algebraic/complex geometry through the theorem of Borel-Weil-Bott, which we then generalize to Beilinson-Bernstein localization
via \(D\)-modules. Familiarity with vector bundles and/or sheaves is recommended.

April 22

Leonardo Abbrescia

The Steamiest Equationin Mathematics

The heat equation is the primary model for all parabolic equations. Thus, in order to understand interesting geometrical problems such as mean curvature flow and Ricci flow,
which are modeled by parabolic equations, we must make understanding the heat equation a priority.
In this talk I will present to you one way to (heuristically) recover the heat equation, prove in detail that this is indeed the solution,
and prove the Weierstrass approximation theorem as an application.
In terms of prerequisites, one should be familiar with calculus at the level presented in Honors Math.

April 29

Xiangwen Zhang

Convex geometryand PDEs

I will start from basic facts about the geometry of convex surfaces and then introduce some important problems, which are both interesting in geometry and PDE.